Leveraging some underlying distance is crucial to define loss functions taking into account the relative positions of the points in the distributions.
This helps to capture the salient features of the problem, since geometrically close samples might share similar properties (such as in domain adaptation [1]}, see Section REF ).
The choice or the design of this distance is thus of primary importance.
In some cases, some natural metric is induced by the problem under study, such as for instance Euclidean or geodesic distance when studying some physical process [2]}, [3]}.
In some other favorable cases, expert knowledge can be leveraged to design a metric, which can be for instance an Euclidean distance on some hand-crafted features. A representative example is in single cell genomics, where the cell can be embedded in a gene's space over which OT can be computed [4]}.
In more complicated setups, such as for images or texts, the metric should be learned in parallel to the resolution of the problem. This “metric learning” setup is still an active area of research [5]}, [6]}, and raises many challenging computational and theoretical questions.
